Question

Difference between $ {n^{th}} $ and $ {(n + 1)^{th}} $ Bohr's radius of $ H $ atom is equal to it's $ {(n - 1)^{th}} $ Bohr's radius. The value of $ n $ is:
 $ (A)1 $
 $ (B)2 $
 $ (C)3 $
 $ (D)4 $

Answer 1

Hint :Bohr radius is a physical constant in atomic physics. It explains the most probable distance between the nucleus and the electron at the ground state of a hydrogen atom. It is named after Niels Bohr, due to its role in the Bohr model of an atom.

Complete Step By Step Answer:
Bohr postulates came forward as the postulates put forward by Rutherford have some drawbacks. The main problem with Rutherford's model was that he couldn't explain why negatively charged electrons remain in orbit when they should instantly fall into the positively charged nucleus.
The drawback of Rutherford’s atomic model lead to the Bohr’s Model where he gave 3 postulates:
(1) First Postulate: The first postulate stated that atoms have some specific stable energy states(known as stationary states) where electrons could orbit around the nucleus without emitting radiation.
(2) Second Postulate: The second postulate state that the orbiting of electrons occur only in the orbits (known as stable orbits) where electrons’ angular momentum $ L $ is equal to the integral multiples of $ \dfrac{h}{{2\pi }} $ , leading to the quantization of moving electron
(3) Third postulate : The third and the last postulate states that when an electron jumps from a higher energy state to a lower energy state, it emit a photon of energy which is equal to the energy difference between the lower and the higher energy states, and its frequency is given by:
 $ hv = {E_i} - {E_f} $
According to the question,
We know that, $ {r_n} \propto {n^2} $
Also, we are given that,
 $ {r_{n + 1}} - {r_n} = {r_{n - 1}} $
 $ {(n + 1)^2} - {n^2} = {(n - 1)^2} $
On further solving, we get,
 $ {n^2} + 1 + 2n - {n^2} = {n^2} + 1 - 2n $
 $ 2n - {n^2} = - 2n $
on taking all the terms on the same side, we get,
 $ {n^2} - 4n = 0 $
 $ n(n - 4) = 0 $
On solving the above equation, we get,
 $ n = 0 $ or $ n = 4 $
Since, $ n = 0 $ is not possible.
So, $ n = 4 $
Hence, the final answer is $ (D)4 $ .

Note :
The Bohr radius is the mean radius of the orbit of an electron around the nucleus of a hydrogen atom at its ground state (lowest-energy level). The value of this radius is a physical constant approximately equal to $ 5.29177 \times {10^{ - 11}}meter $ .